Question

During a rockslide, a \(524-\mathrm{kg}\) rock slides from rest down a hill slope that is \(488 \mathrm{~m}\) long and \(292 \mathrm{~m}\) high. The speed of the rock as it reaches the bottom of the hill is \(62.6 \mathrm{~m} / \mathrm{s}\). How much mechanical energy does the rock lose in the slide due to friction?

Short answer

The rock loses \(4.7 * 10^6 J\) of mechanical energy due to friction.

Step-by-step solution

Calculate initial potential energy

Calculate the initial potential energy that the rock has at the top of the hill using the equation \(PE_i = mgh\), where \(m\) is the mass of the rock, \(g\) is the acceleration due to gravity (approximated as \(9.8 m/s^2\)), and \(h\) is the height of the hill. Plugging the given values this results in \(PE_i = 524 kg * 9.8 m/s^2 * 292 m = 1.5 * 10^7 J\).

Calculate final kinetic energy

Calculate the final kinetic energy that the rock has at the bottom of the hill using the equation \(KE_f = \frac{1}{2}mv^2\), where \(m\) is the mass of the rock and \(v\) is the speed. Plugging the given values this results in \(KE_f = \frac{1}{2} * 524 kg * (62.6 m/s)^2 = 1.03 * 10^7 J\).

Calculate energy loss due to friction

The mechanical energy lost due to friction is the difference between the initial potential energy and the final kinetic energy. Therefore, the energy lost due to friction is calculated as \(E_f = PE_i - KE_f = 1.5 * 10^7 J - 1.03 * 10^7 J = 4.7 * 10^6 J\).

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When considering an object sliding down a hill, like the rock in our exercise, its kinetic energy increases as it accelerates downward. This kind of energy is quantified using the formula:

• \( KE = \frac{1}{2} mv^2 \)

where \(m\) is the mass of the object and \(v\) is its velocity.

In this context, as the rock descends the hill, it transitions from having no kinetic energy (since it starts from rest) to a significant amount as it reaches the bottom with a speed of 62.6 m/s. This is calculated in our exercise as 1.03 \(\times\) 10⁷ Joules at the hill's base.

Understanding kinetic energy helps us grasp how energy changes forms as the rock moves, showcasing the dynamic transformation within mechanical systems.

Potential Energy

Potential energy is the stored energy in an object due to its position or height. For a rock perched atop a hill, potential energy depends on how high it is positioned above the ground, and it’s calculated using the formula:

• \( PE = mgh \)

where \(m\) is the mass, \(g\) is the gravitational acceleration (approximated as 9.8 m/s²), and \(h\) is the height.

The rock in our exercise has an initial potential energy since it starts at rest at the top of a 292-meter high hill. This gives it a potential energy of 1.5 \(\times\) 10⁷ Joules. As the rock slides down, this potential energy is converted into kinetic energy.

Recognizing potential energy is crucial for understanding energy conservation as it highlights how energy is stored and then transformed during movements like sliding down a slope.

Energy Loss Due to Friction

During energy transformations, not all energy is converted perfectly, and some are lost due to frictional forces. Friction is a force that opposes the relative motion of surfaces, converting mechanical energy into heat, causing an energy loss.In the rockslide problem, we can calculate the energy lost to friction by comparing the initial potential energy and the final kinetic energy:

• \( E_{loss} = PE_i - KE_f \)

Plugging in our numbers, the initial potential energy was 1.5 \(\times\) 10⁷ Joules, while the final kinetic energy was 1.03 \(\times\) 10⁷ Joules. The difference, 4.7 \(\times\) 10⁶ Joules, represents the energy lost to friction.

Understanding energy loss is important for appreciating realistic scenarios, where systems are not perfectly efficient, emphasizing the impact of friction in mechanical processes.